By Milgram R. (ed.)

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**Extra resources for Algebraic and Geometric Topology, Part 2, Edition: 1st**

**Example text**

Ti(N −1) (F ) ε for suitable exponents εj = ±1, j = 0, . . , N − 1. We have now seen that the images of F under ΓT cover H. Since the only color and dot preserving isometry of F onto itself is the identity, this means that F is a fundamental domain of ΓT , and ΓT is discrete. For the statement about the hyperbolicity of the elements in ΓT {id} we ﬁrst note that from the fashion in which F is sent to its adjacent neighbors it follows that there is some positive constant δ (depending on the shape of F ) such that d(x, g(x)) ≥ δ for any x ∈ F and any g ∈ ΓT , g = id.

This shows that the only hyperbolic transformations with ﬁxed points 0 and ∞ are the νt ’s. e. if and only if p lies on the axis of νt . Now let z → φ(z) = A[z] with A ∈ SL(2, R) be an arbitrary hyperbolic transformation with ﬁxed points u, w ∈ ∂H. Its axis is γA = C ∩ H, where C is the generalized circle intersecting ∂H orthogonally at u and w. The points u and w are sometimes called the endpoints at inﬁnity of γA . The mapping z → μ(z) = M [z], z ∈ H, M= w w−u −1 w −uw w−u , (34) 1 sends u to 0 and w to ∞.

Tn Isom(H) . Moreover these φ also preserve the coloring and the dotting of the sides. Now if F ∗ is any copy of F in the tiling, then there exists a sequence of copies F0 = F, F1 , F2 , . . , FN −1 , FN = F ∗ in the tiling such that any couple Fj , Fj+1 is a matching pair. From this we conclude that there exists φ ∈ ΓT such that φ(F ) = F ∗ . Indeed, for each I. Hyperbolic Geometry 41 j = 0, . . , N − 1 the common side of the pair Fj , Fj+1 has the color of one of the sides ci(j) of F , and so ε0 ε1 N −1 F ∗ = Ti(0) Ti(1) .